j T' dx = -x^e +EJ ^ 



,2- 



-xVe 



dx=E 



(3) 



Another useful function derivable from equation (1) is the cunnulative 

 distribution function, which gives the probability that an amplitude 



will be less than or equal to the value x. It is given by equation (4) 



F(x) = j -^e d77=l-e (4) 



From this equation, table 1 or any other percentile distribution can be 

 obtained. 



Table 1 stops at 90 percent. One hundred percent of the waves have 

 amplitudes less than infinity, which is all that can be said from 

 equation (1) or equation (4). Theoretically, at least, a wave of very great 

 amplitude is always possible. 



Table 1 



Wave amplitude data in cumulative 

 ascending 10% values 



10% less than 0.32v/e 

 20% less than 0.47^1: 

 30% less than 0.60^1: 

 40% less than 0.71^1: 

 50% less than 0.83Vt; 

 60% less than 0.9(>-/i^ 

 70% less than 1.10^1: 

 80% less than 1.27^1; 

 90% less than 1.5 2^1: 



However, some very ingenious results of Longuet- Higgins (1952) 

 can be used to avoid this difficulty. Longuet- Higgins studied the 

 probability distribution of the highest wave amplitude out of N waves. 

 From this he calculated the average value of the highest wave out 

 of N waves. If a wave record containing a total of NM wave amplitudes 

 is broken up into M pieces of N waves each, and if the M highest 



